Problem: Simplify and expand the following expression: $ \dfrac{k + 8}{4k + 5}+\dfrac{k - 2}{k + 9} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4k + 5)(k + 9)$ Multiply the first term by $\dfrac{k + 9}{k + 9}$ $ \begin{align*} \dfrac{k + 8}{4k + 5} \times \dfrac{k + 9}{k + 9} & = \dfrac{(k + 8)(k + 9)}{(4k + 5)(k + 9)} \\ & = \dfrac{k^2 + 17k + 72}{(4k + 5)(k + 9)}\end{align*} $ Multiply the second term by $\dfrac{4k + 5}{4k + 5}$ $ \begin{align*} \dfrac{k - 2}{k + 9} \times \dfrac{4k + 5}{4k + 5} & = \dfrac{(k - 2)(4k + 5)}{(k + 9)(4k + 5)} \\ & = \dfrac{4k^2 - 3k - 10}{(k + 9)(4k + 5)}\end{align*} $ Now we have: $ = \dfrac{k^2 + 17k + 72}{(4k + 5)(k + 9)} + \dfrac{4k^2 - 3k - 10}{(k + 9)(4k + 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{k^2 + 17k + 72 + 4k^2 - 3k - 10}{(4k + 5)(k + 9)} $ $ = \dfrac{5k^2 + 14k + 62}{(4k + 5)(k + 9)}$ Expand the denominator: $ = \dfrac{5k^2 + 14k + 62}{4k^2 + 41k + 45}$